arXiv subjects
Publications and source records attributed to Bozidar Jovanovic.
Starting from the following multidimensional integrable generalizations of the heavy rigid body systems: the Euler top, the Lagrange top, the Lagrange bitop, and the totally symmetric case, we add to each of them a gyroscope. For each of the newly constructed systems, we provide a polynomial matrix Lax representation and prove Liouville integrability.
We prove the integrability of magnetic geodesic flows of $SO(n)$--invariant Riemannian metrics on the rank two Stefel variety $V_{n,2}$ with respect to the magnetic field $η\, dα$, where $α$ is the standard contact form on $V_{n,2}$ and $η$ is a real parameter. Also, we prove the integrability of magnetic sub-Riemannian geodesic flows for $SO(n)$-invariant sub-Riemannian structures on $V_{n,2}$. All statements in the limit $η=0$ imply the integrability of the problems without the influence of the magnetic field. We also consider integrable pendulum-type natural mechanical systems with the kinetic energy defined by $SO(n)\times SO(2)$--invariant Riemannian metrics. For $n=3$, using the isomorphism $V_{3,2}\cong SO(3)$, the obtained integrable magnetic models reduce to integrable cases of a motion of a heavy rigid body with a gyrostat around a fixed point: Zhukovskiy--Volterra gyrostat, the Lagrange top with a gyrostat, and the Kowalevski top with a gyrostat. As a by-product we obtain the Lax presentations for the Lagrange gyrostat and the Kowalevski gyrostat in the fixed reference frame (dual Lax representations).
There is a natural way to construct sub-Riemannian structures that depend on $n$ parameters on compact Lie groups. These structures are related to the filtrations of Lie subalgebras $\mathfrak g_0 < \mathfrak g_1 < \mathfrak g_2 < \dots < \mathfrak g_{n-1}<\mathfrak g_n=\mathfrak g=Lie(G)$. In the case where $n=1$, the explicit solution for normal sub-Riemannian geodesics was provided by Agrachev, Brockett, and Jurjdevic. We extend their solution to apply to general chains of Lie subgroups. Additionally, we describe normal geodesic lines of the induced sub-Riemannian structures on homogeneous spaces $G/K$, where $\mathfrak g_0=Lie(K)$.
We consider motion of a material point placed in a constant homogeneous magnetic field in $\mathbb R^n$ and also motion restricted to the sphere $S^{n-1}$. While there is an obvious integrability of the magnetic system in $\mathbb R^n$, the integrability of the system restricted to the sphere $S^{n-1}$ is highly non-trivial. We prove complete integrability of the obtained restricted magnetic systems for $n\le 6$. The first integrals of motion of the magnetic flows on the spheres $S^{n-1}$, for $n=5$ and $n=6$, are polynomials of the degree $1$, $2$, and $3$ in momenta. We prove noncommutative integrability of the obtained magnetic flows for any $n\ge 7$ when the systems allow a reduction to the cases with $n\le 6$. We conjecture that the restricted magnetic systems on $S^{n-1}$ are integrable for all $n$.
We formulate the non-commutative integrability of contact systems on a contact manifold $(M,\mathcal H)$ using the Jacobi structure on the space of sections $Γ(L)$ of a contact line bundle $L$. In the cooriented case, if the line bundle is trivial and $\mathcal H$ is the kernel of a globally defined contact form $α$, the Jacobi structure on the space of sections reduces to the standard Jacobi structure on $(M,α)$. We therefore treat contact systems on cooriented and non-cooriented contact manifolds simultaneously. In particular, this allows us to work with dissipative Hamiltonian systems where the Hamiltonian does not have to be preserved by the Reeb vector field.
We study the relativistic formulation of a classical time-dependent nonholonomic Lagrangian mechanics from the perspective of moving frames. We also introduce time-dependent $G$-Chaplygin systems with affine constraints, which are natural objects for the invariant formulation of nonholonomic systems with symmetries. As far as the author is aware, the Hamiltonization problem for time-dependent constraints has not yet been studied. As a first step in this direction, we consider a rolling without sliding of a balanced disc of radius $r$ over a vertical circle of variable radius $R(t)$. We modify the Chaplygin multiplier method and prove that the reduced system becomes the usual Lagrangian system with respect to the new time.
In 1983 Bogoyavlenski conjectured that if the Euler equations on a Lie algebra $\mathfrak g_0$ are integrable, then their certain extensions to semisimple lie algebras $\mathfrak g$ related to the filtrations of Lie algebras $\mathfrak g_0\subset \mathfrak g_1\subset \mathfrak g_2\dots\subset\mathfrak g_{n-1}\subset \mathfrak g_n=\mathfrak g$ are integrable as well. In particular, by taking $\mathfrak g_0=\{0\}$ and natural filtrations of $\mathfrak{so}(n)$ and $\mathfrak{u}(n)$, we have Gel'fand-Cetlin integrable systems. We proved the conjecture for filtrations of compact Lie algebras $\mathfrak g$: the system are integrable in a noncommutative sense by means of polynomial integrals. Various constructions of complete commutative polynomial integrals for the system are also given.
In this note we present invariant formulation of the d'Alambert principle and classical time-dependent Lagrangian mechanics with holonomic constraints from the perspective of moving frames.
We present the basic concepts of space and time, the Galilean and pseudo-Euclidean geometry. We use an elementary geometric framework of affine spaces and groups of affine transformations to illustrate the natural relationship between classical mechanics and theory of relativity, which is quite often hidden, despite its fundamental importance. We have emphasized a passage from the group of Galilean motions to the group of Poincaré transformations of a plane. In particular, a 1-parametric family of natural deformations of the Poincaré group is described. We also visualized the underlying groups of Galilean, Euclidean, and pseudo-Euclidean rotations within the special linear group.
We introduce and study the Chaplygin systems with gyroscopic forces. This natural class of nonholonomic systems has not been treated before. We put a special emphasis on the important subclass of such systems with magnetic forces. The existence of an invariant measure and the problem of Hamiltonization are studied, both within the Lagrangian and the almost-Hamiltonian framework. In addition, we introduce problems of rolling of a ball with the gyroscope without slipping and twisting over a plane and over a sphere in $\mathbb R^n$ as examples of gyroscopic $SO(n)$--Chaplygin systems. We describe an invariant measure and provide examples of $SO(n-2)$--symmetric systems (ball with gyroscope) that allow the Chaplygin Hamiltonization. In the case of additional $SO(2)$--symmetry we prove that the obtained magnetic geodesic flows on the sphere $S^{n-1}$ are integrable. In particular, we introduce the generalized Demchenko case in $\mathbb R^n$, where the inertia operator of the system is proportional to the identity operator. The reduced systems are automatically Hamiltonian and represent the magnetic geodesic flows on the spheres $S^{n-1}$ endowed with the round-sphere metric, under the influence of a homogeneous magnetic field. The magnetic geodesic flow problem on the two-dimensional sphere is well known, but for $n>3$ was not studied before. We perform explicit integrations in elliptic functions of the systems for $n=3$ and $n=4$, and provide the case study of the solutions in both situations.
Motivated by the time-dependent Hamiltonian dynamics, we extend the notion of Arnold-Liouville and noncommutative integrability of Hamiltonian systems on symplectic manifolds to that on cosymplectic manifolds. We prove a variant of the non-commutative integrability for evaluation and Reeb vector fields on cosymplectic manifolds and provide a construction of cosymplectic action-angle variables.
We first construct nonholonomic systems of $n$ homogeneous balls $\mathbf B_1,\dots,\mathbf B_n$ with centers $O_1,...,O_n$ and with the same radius $r$ that are rolling without slipping around a fixed sphere $\mathbf S_0$ with center $O$ and radius $R$. In addition, it is assumed that a dynamically nonsymmetric sphere $\mathbf S$ of radius $R+2r$ and the center that coincides with the center $O$ of the fixed sphere $\mathbf S_0$ rolls without slipping over the moving balls $\mathbf B_1,\dots,\mathbf B_n$. We prove that these systems possess an invariant measure. As the second task, we consider the limit, when the radius $R$ tends to infinity. We obtain a corresponding planar problem consisting of $n$ homogeneous balls $\mathbf B_1,\dots,\mathbf B_n$ with centers $O_1,...,O_n$ and the same radius $r$ that are rolling without slipping over a fixed plane $Σ_0$, and a moving plane $Σ$ that moves without slipping over the homogeneous balls. We prove that this system possesses an invariant measure and that it is integrable in quadratures according to the Euler-Jacobi theorem.
We present an integrable nonholonomic case of rolling without sliding of a gyroscopic ball over a sphere. This case was introduced and studied in detail by Vasilije Demchenko in his 1923 doctoral dissertation defended at the University of Belgrade, with Anton Bilimovic as the advisor. These results are absolutely unknown to modern researchers. The study is based on the C. Neumann coordinates and the Voronec principle. By using involved technique of elliptic functions, a detailed study of motion is performed. Several special classes of trajectories are distinguished, including regular and pseudo-regular precessions. So-called remarkable trajectories, introduced by Paul Painleve and Anton Bilimovic, are described in the present case. The historic context as well as the place of the results in contemporary mechanics are outlined.
We study a time reparametrisation of the Newton type equations on Riemannian manifolds slightly modifying the Chaplygin multiplier method, allowing us to consider the Chaplygin method and the Maupertuis principle within a unified framework. As an example, the reduced nonholonomic problem of rolling without slipping and twisting of an $n$-dimensional balanced ball over a fixed sphere is considered. For a special inertia operator (depending on $n$ parameters) we prove complete integrability when the radius of the ball is twice the radius of the sphere. In the case of $SO(l)\times SO(n-l)$ symmetry, noncommutative integrability for any ratio of the radii is established.
In the review we describe a relation between the Heisenberg spin chain model on pseudospheres and light--like cones in pseudo--Eucli\-dean spaces and virtual billiards. A geometrical interpretation of the integrals associated to a family of confocal quadrics is given, analogous to Moser's geometrical interpretation of the integrals of the Neumann system on the sphere.
We study the rolling of the Chaplygin ball in $\mathbb R^n$ over a fixed $(n-1)$--dimensional sphere without slipping and without slipping and twisting. The problems can be naturally considered within a framework of appropriate modifications of the L+R and LR systems -- well known systems on Lie groups groups with an invariant measure. In the case of the rolling without slipping and twisting, we describe the $SO(n)$-Chaplygin reduction to $S^{n-1}$ and prove the Hamiltonization of the reduced system for a special inertia operator.
We study geometric and algebraic geometric properties of the continuous and discrete Neumann systems on cotangent bundles of Stiefel varieties $V_{n,r}$. The systems are integrable in the non-commutative sense, and by applying a $2r\times 2r$--Lax representation, we show that generic complex invariant manifolds are open subsets of affine (non-compact) Prym varieties on which the complex flow is linear. The characteristics of the varieties and the direction of the flow are calculated explicitly. Next, we construct a family of (multi-valued) integrable discretizations of the Neumann systems and describe them as translations on the Prym varieties, which are written explicitly in terms of divisors of points on the spectral curve. It appears that the systems inherit or naturally generalize the basic properties of the classical Neumann system on $S^{n-1}$ and, therefore, of the Jacobi--Mumford systems: the structure of the Lax matrices, the spectral curve, the equations of motion, linearization on Abelian varieties, and, in the discrete case, the formula for the translation on them.
We consider Noether symmetries of the equations defined by the sections of characteristic line bundles of nondegenerate 1-forms and of the associated perturbed systems. It appears that this framework can be used for time-dependent systems with constraints and nonconservative forces, allowing a quite simple and transparent formulation of the momentum equation and the Noether theorem in their general forms.